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I don't see exactly how your sheet is laid out, but I think it must be something like this:
| A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P |
---|
Project | Start | length | Finish | Quarter end dates | | | | | | | | | | | | |
A | | | | | | | | | | | | | | | | |
B | | | | | | | | | | | | | | | | |
C | | | | | | | | | | | | | | | | |
<tbody>
[TD="align: center"]1[/TD]
[TD="align: right"]3/31/2019[/TD]
[TD="align: right"]6/30/2019[/TD]
[TD="align: right"]9/30/2019[/TD]
[TD="align: right"]12/31/2019[/TD]
[TD="align: right"]3/31/2020[/TD]
[TD="align: right"]6/30/2020[/TD]
[TD="align: right"]9/30/2020[/TD]
[TD="align: right"]12/31/2020[/TD]
[TD="align: right"]3/31/2021[/TD]
[TD="align: right"]6/30/2021[/TD]
[TD="align: right"]9/30/2021[/TD]
[TD="align: center"]2[/TD]
[TD="align: right"]5/11/2019[/TD]
[TD="align: right"]1166[/TD]
[TD="align: right"]7/20/2022[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
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[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]3[/TD]
[TD="align: right"]7/2/2019[/TD]
[TD="align: right"]75[/TD]
[TD="align: right"]9/15/2019[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
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[TD="align: center"]4[/TD]
[TD="align: right"]2/18/2020[/TD]
[TD="align: right"]730[/TD]
[TD="align: right"]2/17/2022[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
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</tbody>
Sheet4
[TABLE="width: 85%"]
<tbody>[TR]
[TD]
Worksheet Formulas[TABLE="width: 100%"]
<tbody>[TR]
[TH]Cell[/TH]
[TH="align: left"]Formula[/TH]
[/TR]
[TR]
[TH]D2[/TH]
[TD="align: left"]=B2+C2[/TD]
[/TR]
</tbody>[/TABLE]
[/TD]
[/TR]
</tbody>[/TABLE]
If that's the case, then select columns F:Z (or however far over your dates go). Click Conditional Formatting > New Rule > Use a Formula > and enter:
=AND(
MIN(F$1,$D1)-MAX(EOMONTH(F$1,-3)+1,$B1)>0,
$A1<>"",
ROW()>1)
Click Format... and select a black fill color.
The first part of the formula (red) is a formula that counts how many days overlap between 2 date ranges. The 2 ranges are B2 and D2, and F1-90 and F1. If the number is >0 then we know that at least part of the project occurred during that quarter. The part in green keeps the formatting from occurring on a blank line, and the part in blue keeps it from highlighting the header line. The absolute symbol ($) is carefully chosen so that it works over the whole range.
If your sheet doesn't look exactly like that and you can't adapt the formula, let me know how it looks and I'll see what I can do.